I'm the non-custodial parent. Do I have to get the custodial parent's permission before cutting my daughter's hair, or letting her pierce her ears?
Strictly speaking, if it isn't in your divorce decree, then no, you don't have to get permission.
Having said that, be prepared to have these kinds of things cause major friction and trouble if you and your ex can't come to some sort of agreement on relatively minor items like haircuts.
Small disputes over things like this often escalate to more serious conflicts, which in turn lead to even more serious conflicts, and so on.
Our advice? Pick your battles wisely. Unless this is genuinely important to you AND the child, ignore it.